arXiv:2410.13362 · The European Physical Journal C

Further evidence for near-Tsirelson Bell–CHSH violations in quantum field theory via Haar wavelets

David Dudal, Ken Vandermeersch

Published Apr 6, 2026• 1 citation

Abstract

Abstract This paper investigates a recent construction using bumpified Haar wavelets to demonstrate explicit violations of the Bell–Clauser–Horne–Shimony–Holt inequality within the vacuum state in Quantum Field Theory. The construction was tested for massless spinor fields in $$(1+1)$$ ( 1 + 1 ) -dimensional Minkowski spacetime and is claimed to achieve violations arbitrarily close to an upper bound known as Tsirelson’s bound. We show that this claim may be reduced to a mathematical conjecture involving the maximal eigenvalue of a sequence of symmetric matrices composed of integrals of Haar wavelet products. More precisely, the asymptotic eigenvalue of this sequence should approach $$\pi $$ π . We present a formal argument using a subclass of wavelets, allowing to reach 3.11052. Although a complete proof remains elusive, we present further compelling numerical evidence to support it.

DOI: 10.1140/epjc/s10052-026-15559-6

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arXiv:2410.13362 · The European Physical Journal C

Further evidence for near-Tsirelson Bell–CHSH violations in quantum field theory via Haar wavelets

David Dudal, Ken Vandermeersch

Published Apr 6, 2026• 1 citation

arXiv HTML PDF

Abstract

(1+1)

( 1 + 1 ) -dimensional Minkowski spacetime and is claimed to achieve violations arbitrarily close to an upper bound known as Tsirelson’s bound. We show that this claim may be reduced to a mathematical conjecture involving the maximal eigenvalue of a sequence of symmetric matrices composed of integrals of Haar wavelet products. More precisely, the asymptotic eigenvalue of this sequence should approach

\pi

π . We present a formal argument using a subclass of wavelets, allowing to reach 3.11052. Although a complete proof remains elusive, we present further compelling numerical evidence to support it.

DOI: 10.1140/epjc/s10052-026-15559-6

Related papers

Maximal violation of the Bell-Clauser-Horne-Shimony-Holt inequality via bumpified Haar wavelets

David Dudal, Philipe De Fabritiis, M. S. Guimarães +2 more

We devise a general setup to investigate the violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality in the vacuum state in the context of quantum field theory. We test the method with massless spinor fields in ( $1+1$ )-dimensional Minkowski space-time. Alice's and Bob's test functions are explicitly constructed, first by employing Haar wavelets which are then bumpified into proper test functions via a smoothening procedure relying on the Planck-taper window function. Relativistic causality is implemented by requiring the support of Alice's and Bob's test functions to be located in the left and right Rindler wedges, respectively. Violations of the Bell-CHSH inequality as close as desired to Tsirelson's bound are reported. We briefly comment on the extra portal, compared to earlier works; this opens the opportunity to scrutinize Bell-CHSH inequalities with generic, interacting quantum field theories.We devise a general setup to investigate the violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality in the vacuum state in the context of quantum field theory. We test the method with massless spinor fields in ($1+1$)-dimensional Minkowski space-time. Alice's and Bob's test functions are explicitly constructed, first by employing Haar wavelets which are then bumpified into proper test functions via a smoothening procedure relying on the Planck-taper window function. Relativistic causality is implemented by requiring the support of Alice's and Bob's test functions to be located in the left and right Rindler wedges, respectively. Violations of the Bell-CHSH inequality as close as desired to Tsirelson's bound are reported. We briefly comment on the extra portal, compared to earlier works; this opens the opportunity to scrutinize Bell-CHSH inequalities with generic, interacting quantum field theories.

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We present an investigation of the Clauser-Horne-Shimony-Holt (CHSH) inequality within a relativistic quantum field theory model built up with a pair of free massive scalar fields $({\ensuremath{\varphi}}_{A},{\ensuremath{\varphi}}_{B})$ where, as is customary, the indices $(A,B)$ refer to Alice and Bob, respectively. A set of bounded Hermitian operators is introduced by making use of the Weyl operators. A CHSH-type correlator is constructed and evaluated in the Fock vacuum by means of canonical quantization. Although the observed violation of the CHSH inequality turns out to be rather small as compared to Tsirelson's bound of quantum mechanics, the model can be employed for the study of Bell's inequalities in the more physical case of gauge theories such as the Higgs models, for which local Becchi-Rouet-Stora-Tyutin (BRST) invariant operators describing both the massive gauge boson as well as the Higgs particle have been devised. These operators can be naturally exponentiated, leading to BRST invariant type of Weyl operators useful to analyze Bell's inequalities within an invariant BRST environment.We present an investigation of the Clauser-Horne-Shimony-Holt (CHSH) inequality within a relativistic quantum field theory model built up with a pair of free massive scalar fields $({\ensuremath{\varphi}}_{A},{\ensuremath{\varphi}}_{B})$ where, as is customary, the indices $(A,B)$ refer to Alice and Bob, respectively. A set of bounded Hermitian operators is introduced by making use of the Weyl operators. A CHSH-type correlator is constructed and evaluated in the Fock vacuum by means of canonical quantization. Although the observed violation of the CHSH inequality turns out to be rather small as compared to Tsirelson's bound of quantum mechanics, the model can be employed for the study of Bell's inequalities in the more physical case of gauge theories such as the Higgs models, for which local Becchi-Rouet-Stora-Tyutin (BRST) invariant operators describing both the massive gauge boson as well as the Higgs particle have been devised. These operators can be naturally exponentiated, leading to BRST invariant type of Weyl operators useful to analyze Bell's inequalities within an invariant BRST environment.

arXiv:2210.16372Physical review. D/Physical review. D.Dec 30, 20229 citations

(1+1)

\pi

π . We present a formal argument using a subclass of wavelets, allowing to reach 3.11052. Although a complete proof remains elusive, we present further compelling numerical evidence to support it.

1+1

)-dimensional Minkowski space-time. Alice's and Bob's test functions are explicitly constructed, first by employing Haar wavelets which are then bumpified into proper test functions via a smoothening procedure relying on the Planck-taper window function. Relativistic causality is implemented by requiring the support of Alice's and Bob's test functions to be located in the left and right Rindler wedges, respectively. Violations of the Bell-CHSH inequality as close as desired to Tsirelson's bound are reported. We briefly comment on the extra portal, compared to earlier works; this opens the opportunity to scrutinize Bell-CHSH inequalities with generic, interacting quantum field theories.

We study Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) violations in the vacuum state of free spinor fields in

(1+1)

-dimensional Minkowski spacetime. We construct explicit smooth compactly supported test functions with spacelike separated supports whose Bell-CHSH correlators converge to Tsirelson's bound

2\sqrt2

. In the massless case, after passage to the time-zero slice and a natural symmetry reduction, the problem reduces to the quadratic form of the Carleman operator on

L^2([0,\infty))

. Near-maximal Bell violation is then governed by the spectral edge

π

, and explicit near-extremizers are obtained from compactly supported cutoffs of the generalized eigenfunction

x^{-1/2}

. This also explains the appearance of the constant

π

in earlier wavelet-based formulations. In the massive case, the same reduction leads to a Hankel operator with kernel

mK_1(m(x+y))

, where

K_1

denotes the modified Bessel function of the second kind of order

1

, and exponentially damped variants of the massless test functions again yield Bell-CHSH values converging to

2\sqrt2

. Therefore, we establish a direct link between Bell-CHSH violations for free

(1+1)

-dimensional spinor fields and the spectral theory of Carleman and Hankel operators on the half-line.

1+1

Minkowski spacetime is scrutinized.

The massive Majorana field in

1+1

dimension is employed to investigate the violation of the Bell-CHSH inequality in relativistic Quantum Field Theory. We give an explicit rapidity-space realization of the Summers-Werner modular-localization construction and reduce the vacuum Bell-CHSH correlator to a single spectral weight

h^2(ω)

for the modular operator. The resulting analytic families approach the Tsirelson bound in the vacuum state as their spectral weight concentrates near

ω\approx0

, corresponding to the eigenvalue

λ^2 \approx 1

of the modular operator.

(A,B)

refer to Alice and Bob, respectively. A set of bounded Hermitian operators is introduced by making use of the Weyl operators. A CHSH-type correlator is constructed and evaluated in the Fock vacuum by means of canonical quantization. Although the observed violation of the CHSH inequality turns out to be rather small as compared to Tsirelson's bound of quantum mechanics, the model can be employed for the study of Bell's inequalities in the more physical case of gauge theories such as the Higgs models, for which local Becchi-Rouet-Stora-Tyutin (BRST) invariant operators describing both the massive gauge boson as well as the Higgs particle have been devised. These operators can be naturally exponentiated, leading to BRST invariant type of Weyl operators useful to analyze Bell's inequalities within an invariant BRST environment.